Problem: The sum of two numbers is $31$, and their difference is $9$. What are the two numbers?
Answer: Let $x$ be the first number, and let $y$ be the second number. The system of equations is: ${x+y = 31}$ ${x-y = 9}$ Solve for $x$ and $y$ using elimination. Add the top and bottom equations together. $ 2x = 40 $ $ x = \dfrac{40}{2} $ ${x = 20}$ Now that you know ${x = 20}$ , plug it back into $ {x+y = 31}$ to find $y$ ${(20)}{ + y = 31}$ ${y = 11}$ You can also plug ${x = 20}$ into $ {x-y = 9}$ and get the same answer for $y$ ${(20)}{ - y = 9}$ ${y = 11}$ Therefore, the larger number is $20$, and the smaller number is $11$.